From the time my son was born to the time we introduced solids, my son had pooping issues. He was formula fed for a while (and was constipated for part of that time), and the I relactated (when he was one and a half months old), and he was breastfed from that point on. He would go 1 or 2 weeks without pooping (which is weird for breastfed babies), but he wasn't constipated, it just wouldn't come out. Once we started feeding him solids, he actually he got regular, and he now poops every single day. Things that make most babies constipated (bananas and rice), help keep him regular. Anyway, you might want to take out all foods and just breastfeed to clean out his system, then slowly try introducing different things again. He doesn't really need the solids yet, so going a few days without won't hurt (just make sure you breastfeed him more if needs it).
I'm pretty sure asparagus made my son a bit constipated, and that's supposed to be something that helps you go. All babies are different, and they react differently to food, which is why you should you sort of play with it. I would go longer on introducing new foods too, to make sure that he isn't getting constipated from that one new food.
You said you're feeding him 3-8 oz twice a day, that is a lot in one sitting for a 7 month old. I started out with 1 or 2 tablespoons once a day, then twice, then added in another feeding. Then once he got to three feedings, started increasing the amount in each meal. I did this very slowly, and it really helped me keep up my milk supply (which is going strong, he's now 11 and a half months old).
One more thing, on him not taking a sippy cup or bottle, I started the sippy cup at about 7 months (no bottles since the first couple of months), and it took him a couple of weeks to get used to it. Don't force it on him, just let him play with it, and help him drink out of it if he seems interested. The other thing you can try is a real cup, my son would drink out of a plastic cup (with help of course), at around 4 months. He would try water when we were drinking it, you might want to try something like that.